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Camera Pose Revisited

Władysław Skarbek, Michał Salomonowicz, Michał Król

TL;DR

This paper revisits camera pose estimation for planar scenes by introducing PnP-ProCay78, a Cayley-parameterized nonlinear LS approach that eliminates translation analytically. It provides a deterministic, two-start initialization based on the kernel structure of the reconstruction-error matrix, aided by a Procrustes$^+$ correction to obtain valid rotations. Empirical results show projection and reprojection accuracy comparable to SQPnP and IPPE while offering a simpler, more interpretable algorithm, with consistent convergence across RGB and low-resolution IR data. The work also contributes a geometric and didactic framework, including Cayley-space trajectory visualization, and discusses implications for multi-sensor (RGB–IR) calibration and future extensions to quasi-planar scenes.

Abstract

Estimating the position and orientation of a camera with respect to an observed scene is one of the central problems in computer vision, particularly in the context of camera calibration and multi-sensor systems. This paper addresses the planar Perspective--$n$--Point problem, with special emphasis on the initial estimation of the pose of a calibration object. As a solution, we propose the \texttt{PnP-ProCay78} algorithm, which combines the classical quadratic formulation of the reconstruction error with a Cayley parameterization of rotations and least-squares optimization. The key component of the method is a deterministic selection of starting points based on an analysis of the reconstruction error for two canonical vectors, allowing costly solution-space search procedures to be avoided. Experimental validation is performed using data acquired also from high-resolution RGB cameras and very low-resolution thermal cameras in an integrated RGB--IR setup. The results demonstrate that the proposed algorithm achieves practically the same projection accuracy as optimal \texttt{SQPnP} and slightly higher than \texttt{IPPE}, both prominent \texttt{PnP-OpenCV} procedures. However, \texttt{PnP-ProCay78} maintains a significantly simpler algorithmic structure. Moreover, the analysis of optimization trajectories in Cayley space provides an intuitive insight into the convergence process, making the method attractive also from a didactic perspective. Unlike existing PnP solvers, the proposed \texttt{PnP-ProCay78} algorithm combines projection error minimization with an analytically eliminated reconstruction-error surrogate for translation, yielding a hybrid cost formulation that is both geometrically transparent and computationally efficient.

Camera Pose Revisited

TL;DR

This paper revisits camera pose estimation for planar scenes by introducing PnP-ProCay78, a Cayley-parameterized nonlinear LS approach that eliminates translation analytically. It provides a deterministic, two-start initialization based on the kernel structure of the reconstruction-error matrix, aided by a Procrustes correction to obtain valid rotations. Empirical results show projection and reprojection accuracy comparable to SQPnP and IPPE while offering a simpler, more interpretable algorithm, with consistent convergence across RGB and low-resolution IR data. The work also contributes a geometric and didactic framework, including Cayley-space trajectory visualization, and discusses implications for multi-sensor (RGB–IR) calibration and future extensions to quasi-planar scenes.

Abstract

Estimating the position and orientation of a camera with respect to an observed scene is one of the central problems in computer vision, particularly in the context of camera calibration and multi-sensor systems. This paper addresses the planar Perspective----Point problem, with special emphasis on the initial estimation of the pose of a calibration object. As a solution, we propose the \texttt{PnP-ProCay78} algorithm, which combines the classical quadratic formulation of the reconstruction error with a Cayley parameterization of rotations and least-squares optimization. The key component of the method is a deterministic selection of starting points based on an analysis of the reconstruction error for two canonical vectors, allowing costly solution-space search procedures to be avoided. Experimental validation is performed using data acquired also from high-resolution RGB cameras and very low-resolution thermal cameras in an integrated RGB--IR setup. The results demonstrate that the proposed algorithm achieves practically the same projection accuracy as optimal \texttt{SQPnP} and slightly higher than \texttt{IPPE}, both prominent \texttt{PnP-OpenCV} procedures. However, \texttt{PnP-ProCay78} maintains a significantly simpler algorithmic structure. Moreover, the analysis of optimization trajectories in Cayley space provides an intuitive insight into the convergence process, making the method attractive also from a didactic perspective. Unlike existing PnP solvers, the proposed \texttt{PnP-ProCay78} algorithm combines projection error minimization with an analytically eliminated reconstruction-error surrogate for translation, yielding a hybrid cost formulation that is both geometrically transparent and computationally efficient.
Paper Structure (24 sections, 3 theorems, 18 equations, 9 figures, 9 tables)

This paper contains 24 sections, 3 theorems, 18 equations, 9 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Related Work
  4. Projective Models in Camera Calibration
  5. Matrix Formulation of the Pinhole Camera Model
  6. Duality Between the Image Plane and the Calibration Plane
  7. Cayley Representation of Rotations
  8. Initial Estimation of the Calibration Board Pose
  9. Experimental Data
  10. The PnP-ProCay78 Algorithm
  11. How Are the Initial Cayley Vectors Selected?
  12. The PnP-ProCay78 Algorithm: Pseudocode
  13. PnP-ProCay78 versus PnP-OpenCV: Experimental Results
  14. Analysis of State Trajectories in Cayley Space
  15. Discussion and Conclusions
  16. ...and 9 more sections

Key Result

Theorem 1

At stationary points with respect to the translation vector $t$, the reconstruction error of perspective projections $p_i\in\mathbb{R}^{3}$, $(p_i)_z=1$, equivalently the reconstruction error on the 3D scene side with respect to points $P_i\in\mathbb{R}^{3}$, given by can be expressed as a quadratic form for a certain matrix $\Omega\in\mathbb{R}^{9\times9}$ with respect to the variable $r\doteq \

Figures (9)

  • Figure S1: Dual camera and calibration board configurations: on the left, the camera is stationary and the board is moving; on the right, the board is stationary and the camera is moving.
  • Figure S2: Mosaic of representative calibration images acquired using five heterogeneous vision and thermovision cameras, two types of boards with chessboard and multiresolution Charuco grids, and both emissive (OLED screen) and reflective light propagation principles (printed calibration board).
  • Figure S3: Views of the RGB and IR cameras used in our experiments.
  • Figure S4: Calibration board views (id=41) displayed sequentially on an OLED screen: on the left, a board with Charuco markers; on the right, a thermal image of a standard chessboard. The chessboard grid perfectly overlaps the Charuco grid at half the resolution. In the second row, trajectories of the LM optimizer states in Cayley space are shown for RGB camera pose estimation (left) and IR camera pose estimation (right), respectively.
  • Figure S5: Trajectories of the TRF optimizer states in Cayley space. The first row corresponds to RGB camera pose estimation, while the second row shows the trajectory for the low-resolution thermal camera.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Theorem 1: On the stationary formula for the quadratic reconstruction error Terzakis2020
  • Theorem 2: On corrective isometry under exchange of camera and board poses
  • Theorem A3: Rotational Procrustes problem